Evaluation of an invariant polynomial on curvature

Let $M$ be a finite-dimensional Hausdorff second-countable smooth manifold, possibly with boundary. Let $\mathbb{F},\mathbb{K}\in \{\mathbb{R},\mathbb{C}\}$, let $G$ be a real or complex matrix Lie group in ${\mathrm{GL}}_r(\mathbb{F})$, and let $\mathfrak{g}$ be its Lie algebra. For a smooth rank-$r$ $\mathbb{F}$-vector bundle $E\to M$, a $G$-frame atlas is a supplied open cover with local frames $e_{\alpha }$ whose transition matrices $A_{\beta \alpha }$, defined by $e_{\beta }=e_{\alpha }{A}_{\beta \alpha }$, take values in $G$. This is the local-frame description of a reduction of the frame group to $G$. If $\mathbb{F}=\mathbb{C}$, a connection means a complex connection as in Complex-linear and metric-compatible bundle connections; for $\mathbb{F}=\mathbb{R}$ it means an ordinary smooth bundle connection. A connection is compatible with this reduction when its connection matrix ${\omega }_{\alpha }$ in every supplied $G$-frame is $\mathfrak{g}$-valued. In the convention of Curvature two-form structure equation, its curvature matrix is

$${\mathrm{\Omega }}_{\alpha }=d{\omega }_{\alpha }+{\omega }_{\alpha }\wedge {\omega }_{\alpha }\mathrm{.}$$

It is $\mathfrak{g}$-valued: for tangent vectors $X,Y$, the second term evaluates to $[{\omega }_{\alpha }(X),{\omega }_{\alpha }(Y)]$, and both this bracket and $d{\omega }_{\alpha }(X,Y)$ lie in $\mathfrak{g}$.

Let $P:\mathfrak{g}\to \mathbb{K}$ be a homogeneous degree-$k$ $G$-invariant polynomial, with symmetric multilinear polarization $P_k$ as in Invariant symmetric polynomials on a matrix Lie algebra. For $k\ge 1$, define its evaluation on curvature in a supplied $G$-frame by the alternating $2k$-form

$$P_k({\mathrm{\Omega }}_{\alpha },\dots ,{\mathrm{\Omega }}_{\alpha })=\sum _{a_1,\dots ,a_k}{P}_k(T_{a_1},\dots ,T_{a_k}){\mathrm{\Omega }}_{\alpha }^{a_1}\wedge \cdots \wedge {\mathrm{\Omega }}_{\alpha }^{a_k},$$

where $(T_a)$ is any basis of $\mathfrak{g}$ and ${\mathrm{\Omega }}_{\alpha }={\sum }_a{T}_a\otimes {\mathrm{\Omega }}_{\alpha }^a$. The right side is the multilinear extension of $P_k$ followed by exterior multiplication, so it is independent of the chosen basis or tensor decomposition. Equivalently, for $v_1,\dots ,v_{2k}\in T_{x}M$,

$$(P_k({\mathrm{\Omega }}_{\alpha }^k){)}_x(v_1,\dots ,v_{2k})=\frac{1}{(2!{)}^k}\sum _{\sigma \in S_{2k}}\mathrm{sgn}(\sigma )P_k\text{\hspace{0.17em}⁣}({\mathrm{\Omega }}_{\alpha }(v_{\sigma (1)},v_{\sigma (2)}),\dots ,{\mathrm{\Omega }}_{\alpha }(v_{\sigma (2k-1)},v_{\sigma (2k)}))\mathrm{.}$$

For $k=0$, set $P_0({\mathrm{\Omega }}_{\alpha }^0)=P_0$, the corresponding constant $\mathbb{K}$-valued $0$-form. A finite sum of homogeneous invariant polynomials is evaluated degree by degree and the resulting forms are added.

These local forms agree on overlaps. Indeed, Vector-bundle curvature is an endomorphism-valued two-form makes the curvature a global $\mathrm{End}(E)$-valued $2$-form. To see its frame transformation, if a fibre vector has coordinate columns $x_{\alpha }=A_{\beta \alpha }{x}_{\beta }$ and an endomorphism has matrices $T_{\alpha },T_{\beta }$, then $T_{\beta }=A_{\beta \alpha }^{-1}{T}_{\alpha }{A}_{\beta \alpha }$. Applying this pointwise to curvature gives ${\mathrm{\Omega }}_{\beta }=A_{\beta \alpha }^{-1}{\mathrm{\Omega }}_{\alpha }{A}_{\beta \alpha }$. Since $A_{\beta \alpha }\in G$, simultaneous $G$-invariance of $P_k$ gives

$$P_k({\mathrm{\Omega }}_{\beta },\dots ,{\mathrm{\Omega }}_{\beta })=P_k({\mathrm{\Omega }}_{\alpha },\dots ,{\mathrm{\Omega }}_{\alpha })\mathrm{.}$$

Thus the local evaluations define a global $\mathbb{K}$-valued $2k$-form. For $\mathbb{K}=\mathbb{C}$, this means a complex-valued form, obtained by complexifying the real form convention; the same local formulas are smooth up to boundary charts by The de Rham complex and pullback extend to manifolds with boundary.

All scalar $2$-form coefficients commute under exterior multiplication because their degree is even. Matrix factors inside polynomial expressions, including traces and determinant coefficients, retain the order prescribed by the matrix polynomial.

The reduction and compatible connection are part of the input when $P$ is invariant only under $G$. For a polynomial invariant under the full general linear group, the construction may use the full frame group. The in-scope applications use ${\mathrm{GL}}_r(\mathbb{C})$-invariant polynomials for Chern forms, their complexified analogues for Pontryagin forms, and the Pfaffian on $\mathfrak{s}\mathfrak{o}(2m)$ with an oriented orthonormal frame for Euler forms.

The local $G$-frames and compatible connection are supplied data. The definition makes no simultaneous global choice of frames, and its construction uses no axiom of choice.